A199949 - OEIS

%I #26 Feb 08 2025 22:59:13

%S 6,5,9,2,6,6,0,4,5,7,6,6,9,4,6,0,7,4,5,3,7,3,4,8,5,7,9,5,6,3,0,6,7,6,

%T 1,1,6,1,5,3,2,8,0,2,1,6,4,4,5,1,6,7,9,7,3,6,0,9,4,5,1,3,0,3,1,4,1,0,

%U 7,3,6,4,4,5,5,8,7,4,2,6,6,2,4,4,0,7,1,9,5,1,9,3,1,6,4,1,4,4,7

%N Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

%C For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).

%C Guide to related sequences, with graphs included in Mathematica programs:

%C a.... b.... c.... least x, greatest x

%C 1.... 1.... 2.... A199949, A199950

%C 1.... 1.... 3.... A199951, A199952

%C 1.... 1.... 4.... A199953, A199954

%C 1.... 2.... 3.... A199955, A199956

%C 1.... 2.... 4.... A199957, A199958

%C 1.... 3.... 3.... A199959, A199960

%C 1.... 3.... 4.... A199961, A199962

%C 1.... 4.... 3.... A199963, A199964

%C 1.... 4.... 4.... A199965, A199966

%C 2.... 1.... 3.... A199967, A200003

%C 2.... 1.... 4.... A200004, A200005

%C 3.... 1.... 4.... A200006, A200007

%C 4.... 1.... 4.... A200008, A200009

%C 1... -1.... 1.... A200010, A200011

%C 1... -1.... 2.... A200012, A200013

%C 1... -1.... 3.... A200014, A200015

%C 1... -1.... 4.... A200016, A200017

%C 1... -2.... 1.... A200018, A200019

%C 1... -2.... 2.... A200020, A200021

%C 1... -2.... 3.... A200022, A200023

%C 1... -2.... 4.... A200024, A200025

%C 1... -3.... 1.... A200026, A200027

%C 1... -3.... 2.... A200093, A200094

%C 1... -3.... 3.... A200095, A200096

%C 1... -3.... 4.... A200097, A200098

%C 1... -4.... 1.... A200099, A200100

%C 1... -4.... 2.... A200101, A200102

%C 1... -4.... 3.... A200103, A200104

%C 1... -4.... 4.... A200105, A200106

%C 2... -1.... 1.... A200107, A200108

%C 2... -1.... 2.... A200109, A200110

%C 2... -1.... 3.... A200111, A200112

%C 2... -1.... 4.... A200114, A200115

%C 2... -2.... 1.... A200116, A200117

%C 2... -2.... 3.... A200118, A200119

%C 2... -3.... 1.... A200120, A200121

%C 2... -3.... 2.... A200122, A200123

%C 2... -3.... 3.... A200124, A200125

%C 2... -3.... 4.... A200126, A200127

%C 2... -4.... 1.... A200128, A200129

%C 2... -4.... 3.... A200130, A200131

%C 3... -1.... 1.... A200132, A200133

%C 3... -1.... 2.... A200223, A200224

%C 3... -1.... 3.... A200225, A200226

%C 3... -1.... 4.... A200227, A200228

%C 3... -2.... 1.... A200229, A200230

%C 3... -2.... 2.... A200231, A200232

%C 3... -2.... 3.... A200233, A200234

%C 3... -2.... 4.... A200235, A200236

%C 3... -3.... 1.... A200237, A200238

%C 3... -3.... 2.... A200239, A200240

%C 3... -3.... 4.... A200241, A200242

%C 3... -4.... 1.... A200277, A200278

%C 3... -4.... 2.... A200279, A200280

%C 3... -4.... 3.... A200281, A200282

%C 3... -4.... 4.... A200283, A200284

%C 4... -1.... 1.... A200285, A200286

%C 4... -1.... 2.... A200287, A200288

%C 4... -1.... 3.... A200289, A200290

%C 4... -1.... 4.... A200291, A200292

%C 4... -2.... 1.... A200293, A200294

%C 4... -2.... 3.... A200295, A200296

%C 4... -3.... 1.... A200299, A200300

%C 4... -3.... 2.... A200297, A200298

%C 4... -3.... 3.... A200301, A200302

%C 4... -3.... 4.... A200303, A200304

%C 4... -4.... 1.... A200305, A200306

%C 4... -4.... 3.... A200307, A200308

%C Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.

%C For an example related to A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

%H G. C. Greubel, <a href="/A199949/b199949.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e least x:  0.659266045766946074537348579563067611...

%e greatest x: 1.2710268008159460640047188480978502...

%t (* Program 1:  A199949 *)

%t a = 1; b = 1; c = 2;

%t f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]

%t Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]

%t RealDigits[r]   (* A199949 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]

%t RealDigits[r]   (* A199950 *)

%t (* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)

%t f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];

%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];

%t ListPlot3D[Flatten[t, 1]]  (* for A199949 *)

%o (PARI) a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 05 2018

%Y Cf. A199950.

%K nonn,cons,changed

%O 0,1

%A _Clark Kimberling_, Nov 12 2011

%E A-number corrected by _Jaroslav Krizek_, Nov 27 2011